By approximately matching the information given in the three columns of the following table.

Let f(x) = x + log_e x $$-$$ x log_e x, x$$ \in $$(0, $$\infty $$)

Column 1 contains information about zeroes of f(x), f'(x) and f"(x).

Column 2 contains information about the limiting behaviour of f(x), f'(x) and f"(x) at infinity.

Column 3 contains information about increasing/decreasing nature of f(x) and f'(x).

	Column - 1	Column - 2	Column - 3
(i)	f(x) = 0 for some $$x \in (1,{e^2})$$	(i) $$\mathop {\lim }\limits_{x \to \infty } \,f(x) = 0$$	f is increasing in (0, 1)
(ii)	f'(x) = 0 for some $$x \in (1,e)$$	$$\mathop {\lim }\limits_{x \to \infty } \,f(x) = - \infty $$	f is decreasing in (e, $${e^2}$$)
(iii)	f'(x) = 0 for some $$x \in (0,1)$$	$$\mathop {\lim }\limits_{x \to \infty } \,f'(x) = - \infty $$	f' is increasing in (0, 1)
(iv)	f'(x) = 0 for some $$x \in (1,e)$$	$$\mathop {\lim }\limits_{x \to \infty } \,f'(x) = 0$$	f' is decreasing in (e, $${e^2}$$)

Which of the following options is the only CORRECT combination?

By approximately matching the information given in the three columns of the following table.

Let f(x) = x + log_e x $$-$$ x log_e x, x$$ \in $$(0, $$\infty $$)

Column 1 contains information about zeroes of f(x), f'(x) and f"(x).

Column 2 contains information about the limiting behaviour of f(x), f'(x) and f"(x) at infinity.

Column 3 contains information about increasing/decreasing nature of f(x) and f'(x).

	Column - 1	Column - 2	Column - 3
(i)	f(x) = 0 for some $$x \in (1,{e^2})$$	(i) $$\mathop {\lim }\limits_{x \to \infty } \,f(x) = 0$$	f is increasing in (0, 1)
(ii)	f'(x) = 0 for some $$x \in (1,e)$$	$$\mathop {\lim }\limits_{x \to \infty } \,f(x) = - \infty $$	f is decreasing in (e, $${e^2}$$)
(iii)	f'(x) = 0 for some $$x \in (0,1)$$	$$\mathop {\lim }\limits_{x \to \infty } \,f'(x) = - \infty $$	f' is increasing in (0, 1)
(iv)	f'(x) = 0 for some $$x \in (1,e)$$	$$\mathop {\lim }\limits_{x \to \infty } \,f'(x) = 0$$	f' is decreasing in (e, $${e^2}$$)

Which of the following options is the only CORRECT combination?

The least value of a $$ \in R$$ for which $$4a{x^2} + {1 \over x} \ge 1,$$, for all $$x>0$$. is

Let $$f:\left[ {0,1} \right] \to R$$ (the set of all real numbers) be a function. Suppose the function $$f$$ is twice differentiable,
$$f(0) = f(1)=0$$ and satisfies $$f''\left( x \right) - 2f'\left( x \right) + f\left( x \right) \ge .{e^x},x \in \left[ {0,1} \right]$$.

Which of the following is true for $$0 < x < 1?$$